Question

please don't spam and answer with solution ​

Answer 1

Step-by-step explanation:

31.

Consider the given number

$(1 + a)(1 + b)(1 + c) = (1 + a + b + ab)(1 + c) = (1 + a + b + c+ ab +bc + ac + abc)$

Now,

$[(1 + a)(1 + b)(1 + c)]^{7} = [1 + a + b + c + ab + bc + ca + abc)]^{7} > [a + b + c + ab + bc + ca + abc]^{7} ...(i)$

We know that, for any numbers, its AM ≥ GM

$\frac{a + b + c + ab + bc + ca + abc}{7} \geqslant (a.b.c.ab.bc.ca.abc)^{ \frac{1}{7} }$

$\implies a + b + c + ab + bc + ca + abc \geqslant 7.( {a}^{4} {b}^{4} {c}^{4} ) ^{ \frac{1}{7} }$

$\implies( a + b + c + ab + bc + ca + abc )^{7} \geqslant 7^{7} .( {a}^{4} {b}^{4} {c}^{4} )$

Now, from (i) we get,

$(1 + a + b + c + ab + bc + ca + abc)^{7} > {7}^{7} . {a}^{4} {b}^{4} {c}^{4}$